What are the variables used in a Gaussian radial basis kernel in the context of SVMs?
$\mathbf{x} \in \mathbb{R}^p$ and $\mathbf{x}' \in \mathbb{R}^p$ are two inputs (or feature vectors). 

In the context of classification with an SVM, you are given a dataset $D = \{(\mathbf{x}_i, y_i) \}_{i=1}^N$, where $\mathbf{x}_i \in \mathbb{R}^p$ is an input (or point) and $y_i$ the corresponding label. The goal is to find a hyperplane that classifies the points $\mathbf{x}_i$. The hyperplane actually corresponds to a binary classifier that splits the plane into two, so the assumption is that there are two labels. However, these points $ \mathbf{x}_i$ may not be linearly separable in $\mathbb{R}^p$, i.e. there may not be a hyperplane (in 2d, i.e. when $p=2$, a hyperplane is a line) that separates them. The kernel trick, i.e. the use of kernels (such as the Gaussian radial basis), allows an SVM to perform non-linear classification by transforming the inputs to a space where they are linearly separable.